The forward displacement of the wheel is equal to the linear displacement of a point fixed on the rim. When the angular velocity is expressed with respect to a coordinate system coinciding with the principal axes of the body, each component of the angular momentum is a product of a moment of inertia a principal value of the inertia tensor times the corresponding component of the angular velocity; the torque is the inertia tensor times the angular acceleration.

Like the approximation of a rigid body as a particle, this is never strictly true. The relative terms are the velocity or acceleration measured by an observer attached to the moving reference at particle B. Figure A cylindrical pair C-pair A cylindrical pair keeps two axes of two rigid bodies aligned.

In the following, we will restrict attention to the planar motion of rigid bodies.

For example, when a figure skater pulls in her extended arms, her moment of inertia will decrease, causing an increase in angular velocity.

Decomposition of rigid body motion. The direction of the velocity is tangent to the path of the point of rotation. Consequently, we have the property that all lines on a rigid body in its plane of motion have the same angular displacement, the same angular velocity and the same angular acceleration.

The quantity mr 2 is defined as moment of inertia of a point mass about the center of rotation. Angular velocity and angular acceleration The angular displacement of a rotating wheel is the angle between the radius at the beginning and the end of a given time interval. The total kinetic energy is simply the sum of translational and rotational energy.

Consider two particles A and B moving along independent trajectories in the plane, and a fixed reference O. Even though the masses of the two objects are equal, it is intuitive that the flywheel will be more difficult to push to a high number of revolutions per second because not only the amount of mass but also the distribution of the mass affects the ease in initiating rotation for a rigid body.

Imagine two objects of the same mass with different distribution of that mass. Observe that, as the moving frame does not rotate, basis vectors i and j do not change in time.

A rotation represented by an Euler axis and angle. Here, the rigid body is rotated about through and then translated along the screw axis by an amount. However, depending on the application, a convenient choice may be: The motion of the body is completely determined by the angular velocity of the rotation.

At any time it is equal to the total mass of the rigid body times the translational velocity. Next, consider the motion of a rigid body over the interval as shown, with arbitrary point taken as reference. Rotation about a fixed axis: In the following, we will restrict attention to the planar motion of rigid bodies.

Specifically, inEuler showed that the motion of a body that satisfies 2 is such that 3 where is a rotation tensor. All bodies deform as they move. Two rigid bodies constrained by a revolute pair have an independent rotary motion around their common axis.

Calculating the degrees of freedom of a rigid body system is straight forward.

Another alternative, synonymous with a famous theorem credited to Michel Chasles — [ 3 ], represents the motion of a rigid body by a screw motion.

A rigid body is an object with a mass that holds a rigid shape, such as a phonograph turntable, in contrast to the sun, which is a ball of gas. Two possible choices of the screw axis intercept are also shown.

These are related to the intermediate angle shown as, Observe that as the body is rigid, requiring that the distance between each pair of points on the two lines 1 and 2 is constant, angle must be invariant. To discuss an alternative to 7 that does not feature the reference configurationwe consider the motion of the body during a time intervalas illustrated in Figure 2.Introduction to Mechanisms.

Yi Zhang with Susan Finger Stephannie Behrens Table of Contents. 4 Basic Kinematics of Constrained Rigid Bodies Degrees of Freedom of a Rigid Body.

Degrees of Freedom of a Rigid Body in a Plane. The degrees of freedom (DOF) of a rigid body is defined as the number of independent movements it has. Figure shows a rigid body in a plane. Kinematics Linear and angular position. The position of a rigid body is the position of all the particles of which it is composed.

To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. Introduction to Mechanisms. Yi Zhang with Susan Finger Stephannie Behrens Table of Contents.

4 Basic Kinematics of Constrained Rigid Bodies Degrees of Freedom of a Rigid Body. Degrees of Freedom of a Rigid Body in a Plane. The degrees of freedom (DOF) of a rigid body is defined as the number of independent movements it has.

Figure shows a rigid body. PLANAR KINEMATICS OF A RIGID BODY I. Rigid body motion: As a result, the kinematics of particle motion discussed in chapter 2, may also be used to specify the kinematics of points located in a translating rigid body. II. Rotation about a ﬁxed axis. Plane Kinematics of Rigid Bodies Rigid Body • A system of particles for which the distances between the particles remain unchanged.

• This is an ideal case. CHAPTER 2. KINEMATICS 23 Kinematics of a rigid body The description of motion isrelative. Any velocity or acceleration is expressed with .

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